ARPACK is a Fortran package which provides routines for quickly finding a few
eigenvalues/eigenvectors of large sparse matrices. In order to find these
solutions, it requires only left-multiplication by the matrix in question.
This operation is performed through a reverse-communication interface. The
result of this structure is that ARPACK is able to find eigenvalues and
eigenvectors of any linear function mapping a vector to a vector.

All of the functionality provided in ARPACK is contained within the two
high-level interfaces scipy.sparse.linalg.eigs and
scipy.sparse.linalg.eigsh. eigs
provides interfaces to find the
eigenvalues/vectors of real or complex nonsymmetric square matrices, while
eigsh provides interfaces for real-symmetric or complex-hermitian
matrices.

Note that ARPACK is generally better at finding extremal eigenvalues: that
is, eigenvalues with large magnitudes. In particular, using which='SM'
may lead to slow execution time and/or anomalous results. A better approach
is to use shift-invert mode.

Imagine you’d like to find the smallest and largest eigenvalues and the
corresponding eigenvectors for a large matrix. ARPACK can handle many
forms of input: dense matrices such as numpy.ndarray instances, sparse
matrices such as scipy.sparse.csr_matrix, or a general linear operator
derived from scipy.sparse.linalg.LinearOperator. For this example, for
simplicity, we’ll construct a symmetric, positive-definite matrix.

We now have a symmetric matrix X with which to test the routines. First
compute a standard eigenvalue decomposition using eigh:

>>> evals_all,evecs_all=eigh(X)

As the dimension of X grows, this routine becomes very slow. Especially
if only a few eigenvectors and eigenvalues are needed, ARPACK can be a
better option. First let’s compute the largest eigenvalues (which='LM')
of X and compare them to the known results:

The results are as expected. ARPACK recovers the desired eigenvalues, and they
match the previously known results. Furthermore, the eigenvectors are
orthogonal, as we’d expect. Now let’s attempt to solve for the eigenvalues
with smallest magnitude:

Oops. We see that as mentioned above, ARPACK is not quite as adept at
finding small eigenvalues. There are a few ways this problem can be
addressed. We could increase the tolerance (tol) to lead to faster
convergence:

We get the results we’d hoped for, but the computation time is much longer.
Fortunately, ARPACK contains a mode that allows quick determination of
non-external eigenvalues: shift-invert mode. As mentioned above, this
mode involves transforming the eigenvalue problem to an equivalent problem
with different eigenvalues. In this case, we hope to find eigenvalues near
zero, so we’ll choose sigma=0. The transformed eigenvalues will
then satisfy , so our
small eigenvalues become large eigenvalues .

We get the results we were hoping for, with much less computational time.
Note that the transformation from takes place
entirely in the background. The user need not worry about the details.

The shift-invert mode provides more than just a fast way to obtain a few
small eigenvalues. Say you
desire to find internal eigenvalues and eigenvectors, e.g. those nearest to
. Simply set sigma=1 and ARPACK takes care of
the rest:

The eigenvalues come out in a different order, but they’re all there.
Note that the shift-invert mode requires the internal solution of a matrix
inverse. This is taken care of automatically by eigsh and eigs,
but the operation can also be specified by the user. See the docstring of
scipy.sparse.linalg.eigsh and
scipy.sparse.linalg.eigs for details.